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This is very helpful!

Basically what I take away is that we're inventing a new number system from scratch. So we're not "proving" that 1 is a number between 0 and the empty set. We're defining it as such, and it just so happens that a number system defined this way works out in convergent ways with other mathematics.

Is that roughly right?

Exactly.

There has to be some procedure how to come up with "new" numbers though, if you want to have more in the end than just a fancy binary tree - in particular if you want to map your "fake numbers" to the reals, infinity, etc.

This procedure is enough. If you define addition and other operations in a certain way (as Conway did), it turns out that on the omega-th day (i.e. after initial infinite steps), all reals will be born.

> position does matter

Only as a mental abstraction that's based on our experience/concept of space+time.